# Closure of Subset of Closed Set of Topological Space is Subset/Proof 1

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## Theorem

Let $T$ = $\struct {S, \tau}$ be a topological space.

Let $F$ be a closed set of $T$.

Let $H \subseteq F$ be a subset of $F$.

Let $H^-$ denote the closure of $H$.

Then $H^- \subseteq F$.

## Proof

\(\displaystyle H\) | \(\subseteq\) | \(\displaystyle F\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle H^-\) | \(\subseteq\) | \(\displaystyle F^-\) | Topological Closure of Subset is Subset of Topological Closure | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle F\) | Set is Closed iff Equals Topological Closure |

$\blacksquare$